ARIADNE is a nuclear-magnetic-resonance-based experiment that will search for novel axion-induced spin-dependent interactions between an unpolarized source mass rotor and a nearby sample of spin-polarized $^3$He gas. To detect feeble axion signals at the sub-atto-Tesla level, the experiment relies on low magnetic background and noise. We measure and characterize the magnetic field background from a prototype tungsten rotor. We show that the requirement is met with our current level of tungsten purity and demagnetization process. We further show that the noise is dominantly caused by a few discrete dipoles, likely due to a few impurities trapped inside the rotor during manufacturing. This is done via a numerical optimization pipeline which fits for the locations and magnetic moments of each dipole. We find that under the current demagnetization, the magnetic moment of trapped impurities is bounded at $10^{-9} mathrm{A}mathrm{m}^2$.
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